Low space computation and branching program

Question

One of the most elemental result of relationship between boolean circuit size and polynomial uniform computation is Pippenger and Fishers simulation: $DTIME[T(n)]\subseteq SIZE[T(n)\log T(n)]$.

I want to consider the small branching program family simulating low space computation.

Question1: What is the best simulation bound $S_{1}(n)$ which is known at this moment such that $DSPACE [S(n)]\subseteq BP-SIZE[S_{1}(n)]$

Question2:What is the best time complexity bound as far as we know, for the function that $1^{n}\rightarrow $a branching program in the Question 1.

Answer 1

Denoting the class of functions computed by branching programs of size $f(n)$ by $\text{BP}(f)$, the best known bound seems to be the trivial $\text{DSPACE}(S(n)) \subseteq \text{BP}(2^{O(S(n))})$. This matches the known inclusions $\text{DSPACE}(\log n) = \text{L} \subseteq \text{L}/poly = \text{BP}(poly)$; if a better bound were known then I would expect it to have been noted by Razborov in his FCT 1991 survey (preprint), or in Jukna's 2012 BFC textbook.